250 XIV INTEGRATION IN LOCALLY COMPACT GROUPS (b) Suppose that 0 e A n B. A pair (A', B') of integrable subsets of G is said to be derived from (A, B) if there exists a sequence (sk}i^k$n of elements of G and two se- quences (AJo^&^f, and (BjOo^fc*;,, of subsets of G such that for 1 <; k < n, and sk e A*-i for 1 <J k g /?, and A7 = An , B' = Bn . Show that there exists a sequence (En, Fn) of pairs of subsets of G such that (i) E0 = A and F0 = B, (ii) (En+j,Fn + 1) is derived from (En,Ffl), (iii) p.((EH - s) n Fn) ^ ju(FB+1) - 2~n for all n and all j 6 En . Let EOO = (J En , F« = f| F« • Snow that for each s e Ew we have - s) n Fa,) = /*(F«>). (c) Suppose that /i(F«,) > 0. Show that the function /O) = ^((£,0 - s) n Fco) takes only the values 0 and ft(Foo). Let C be the set of elements s e G such that /O) = /x(Foo). Show that C is open and closed, that /x(C) = /i(Ew) and that C is the closure of Ex (use Problems 2(a) and 2(b)). Let D be the set of all s e Pm such that the intersection of Foo with every neighborhood of s has measure >0. Show that /z(D) = /A(Foo) and that £« -f- D c c. Deduce that D is contained in the subgroup H(C) defined in Problem 3, and that H(C) is a compact open subgroup of G. Show also that C + H(C) = C, that /x(C) ^ ft(A) + /*(B) - /x(H(C)) and that C c A + B (consider the measure of £«, n (c — F«,) for each c e C). (d) Let A, B be two integrable subsets of G. Deduce from (c) that either /-t*(A 4- B)§: ft(A) -f- ju-(B), or there exists a compact open subgroup H in G such that A + B contains a coset of H, and that in this case /4*(A -f B) ^ /x(A) 4- /^(B) — / Consider the case when G is connected. 5. (a) Let A (resp. B) be the set of real numbers x = XQ + '£xi2~t where x0 is an t = i integer, each xt (/ ^ 1) is 0 or 1 , and xt = 0 for all even / > 0 (resp. Xi = 0 for all odd / > 0). Show that each of A, B has zero Lebesgue measure but that A -f- B = R. (b) Deduce from (a) that there exists a Hamel basis H of R (over Q) contained in A u B and therefore of measure zero. The set Pj. of numbers of the form rh, where r 6 Q and h e H, is also of measure 0. (c) Let Pn denote the set of real numbers which have at most n nonzero coordinates relative to the basis H. Show that if Pn is negligible and Pn+I Lebesgue-measurable, then Pw + 1 is negligible. (Let h0 e H, and show that the set S of numbers x e Pn+1 in which the coefficient of h0 is nonzero, is negligible. Using Problem 2(c), show that if Pn+l were not negligible there would exist two points x', x" in Pn + i n OS such that (x' — x")/hQ were rational, and hence obtain a contradiction.) (d) Deduce from (b) and (c) that there exist two negligible sets C, D in R such that C + D is not measurable (with respect to Lebesgue measure). 6. Let G be a group acting (on the left) on a set X. A subset P (resp. C) of X is said to be a G-packing (resp. a G-covering} if for each s ^ e in G we have s • P r\ P = 0 (resp. if X = (J s • C). A subset P which is both a G-packing and a G-covering is called a 56 G G-tesselfation. (a) Suppose that X is separable, metrizable and locally compact, that G is at most denumerable and acts continuously on X (with respect to the discrete topology on G)nuous on the right, defined by w(t) — u(t)v(t) for / J> 0 and w(r) = 0 for / < 0.